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Full text of "Treatise On Analysis Vol-Ii"

2 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA A topological space is a set E endowed with a topology O. The subsets of E which belong to D are called the open sets of the topological space E. Taking L = 0 in (Oj), it follows that the empty subset 0 of E is always an open set, and from (On) we see that E itself is an open set. Examples (12.1.1) The set O of open sets in a metric space E satisfies (Oi) and (On) (3.5.2 and 3.5.3). This topology O is called the topology of the metric space E (or is said to be defined by the distance given on E), Two topologically equiva- lent distances (3.12) define the same topology. A topological space is said to be metrizable if its topology can be defined by a distance (and then this topology is also said to be metrizable). On any set E, the set O = {0, E} is a topology, called the chaotic topology. It is not metrizable if E has at least two elements, because otherwise there would exist an open ball containing one of these elements and not the other, and this is impossible because E is the only nonempty open set. On a set E = {a, b] consisting of two elements, the set £> = (0, {a}, E} is a nonmetrizable topology. If OJL, €>2 are two topologies on the same set E, we say that O2 is finer than £>! (or that C^ is coarser than O2) if Oj c: O2. Two topologies on E are said to be comparable if one is finer than the other. The chaotic topology is coarser than all others. The discrete topology (i.e., the topology defined by the metric (3.2.5), for which O = ^J(E)) is finer than all others. Two topologies on E are not necessarily comparable; for example, let E = {a, b} be a set with two elements, and consider the two topologies D1 = (0, {a}, E} and 02 = (0, {b}, E}. 2. TOPOLOGICAL CONCEPT% We have already remarked (3.12), in the context of metric spaces, that the notions of closed set, cluster point of a set, interior point of a set, frontier point of a set, neighborhood of a point (or of a set), dense set, continuous function and homeomorphism are defined entirely in terms of the notion of an open set in a metric space. Their definitions can therefore be transferred, without any change, to the context of arbitrary topological spaces. Moreover, all the properties involving these notions which were proved for metric spaces, and whose statements do< not involve the distance (cf. (3.5) to (3.12)) remain valid for arbitrary topological spaces (and we shall therefore make use of them in the general case) with the following exceptions:ar operator: 15.11,